Harmony of 23edo: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-30 13:11:45 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-30 13:25:49 UTC</tt>.<br>

: The original revision id was <tt>~~79928729~~</tt>.<br>

: The original revision id was <tt>79930653</tt>.<br>

: The revision comment was: <tt>~~listed 10 triads, 6 tetrads, & one quintad in 23edo~~</tt><br>

: The revision comment was: <tt>added inversions of chords</tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

<h4>Original Wikitext content:</h4>

Line 24:

I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Thus we produce ten triads, five tetrads, & one quintad.

Thus we produce ten triads, five tetrads, & one quintad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.

==Triads==

===16:17:18, degrees 0, 2, 4.===

===16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).===

17/16 (104.955, error -.607)

18/16 = 9/8 (203.910, error +4.786)

18/17 (98.955, error: +5.393)

===16:17:21, degrees 0, 2, 9.===

===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).===

17/16 (104.955, error -.607)

21/16 (470.781, error -1.216)

21/17 (365.825, error: -.608)

===16:17:23, degrees 0, 2, 12.===

===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).===

17/16 (104.955, error -.607)

23/16 (628.274, error -2.187)

23/17 (523.319, error: -1.578)

===16:18:21, degrees 0, 4, 9.===

===16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).===

18/16 = 9/8 (203.910, error +4.786)

21/16 (470.781, error -1.216)

21/18 = 7/6 (266.871, error: -6.001)

===16:18:23, degrees 0, 4, 12.===

===16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).===

18/16 = 9/8 (203.910, error +4.786)

23/16 (628.274, error -2.187)

23/18 (424.364, error: -6.973)

===16:21:23, degrees 0, 9, 12.===

===16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).===

21/16 (470.781, error -1.216)

23/16 (628.274, error -2.187)

23/21 (157.493, error: -.971)

===17:18:21, degrees 0, 2, 7.===

===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).===

18/17 (98.955, error: +5.393)

21/17 (365.825, error: -.608)

21/18 = 7/6 (266.871, error: -6.001)

===17:18:23, degrees 0, 2, 10.===

===17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).===

18/17 (98.955, error: +5.393)

23/17 (523.319, error: -1.578)

23/18 (424.364, error: -6.973)

===17:21:23, degrees 0, 7, 10.===

===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).===

21/17 (365.825, error: -.608)

23/17 (523.319, error: -1.578)

23/21 (157.493, error: -.971)

===18:21:23, degrees 0, 5, 8.===

===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).===

21/18 = 7/6 (266.871, error: -6.001)

23/18 (424.364, error: -6.973)

Line 80:

==Tetrads==

===16:17:18:21, degrees 0, 2, 4, 9.===

===16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).===

17/16 (104.955, error -.607)

18/16 = 9/8 (203.910, error +4.786)

Line 88:

21/18 = 7/6 (266.871, error: -6.001)

===16:17:18:23, degrees 0, 2, 4, 12.===

===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).===

17/16 (104.955, error -.607)

18/16 = 9/8 (203.910, error +4.786)

Line 96:

23/18 (424.364, error: -6.973)

===16:17:21:23, degrees 0, 2, 9, 12.===

===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).===

17/16 (104.955, error -.607)

21/16 (470.781, error -1.216)

Line 104:

23/21 (157.493, error: -.971)

===16:18:21:23, degrees 0, 4, 9, 12.===

===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).===

18/16 = 9/8 (203.910, error +4.786)

21/16 (470.781, error -1.216)

Line 112:

23/21 (157.493, error: -.971)

===17:18:21:23, degrees 0, 2, 7, 10.===

===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).===

18/17 (98.955, error: +5.393)

21/17 (365.825, error: -.608)

Line 122:

==Quintad==

===16:17:18:21:23, degrees 0, 2, 4, 9, 12.===

===16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).===

17/16 (104.955, error -.607)

18/16 = 9/8 (203.910, error +4.786)

Line 279:

I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp; 23, excluding (at least for now) 33, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.<br />

<br />

Thus we produce ten triads, five tetrads, &amp; one quintad.<br />

Thus we produce ten triads, five tetrads, &amp; one quintad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.<br />

<br />

<h2 id="toc0"><a name="x-Triads"></a>Triads</h2>

<br />

<h3 id="toc1"><a name="x-Triads-16:17:18, degrees 0, 2, 4."></a>16:17:18, degrees 0, 2, 4.</h3>

<h3 id="toc1"><a name="x-Triads-16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)."></a>16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).</h3>

17/16 (104.955, error -.607)<br />

18/16 = 9/8 (203.910, error +4.786)<br />

18/17 (98.955, error: +5.393)<br />

<br />

<h3 id="toc2"><a name="x-Triads-16:17:21, degrees 0, 2, 9."></a>16:17:21, degrees 0, 2, 9.</h3>

<h3 id="toc2"><a name="x-Triads-16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21)."></a>16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).</h3>

17/16 (104.955, error -.607)<br />

21/16 (470.781, error -1.216)<br />

21/17 (365.825, error: -.608)<br />

<br />

<h3 id="toc3"><a name="x-Triads-16:17:23, degrees 0, 2, 12."></a>16:17:23, degrees 0, 2, 12.</h3>

<h3 id="toc3"><a name="x-Triads-16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21)."></a>16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).</h3>

17/16 (104.955, error -.607)<br />

23/16 (628.274, error -2.187)<br />

23/17 (523.319, error: -1.578)<br />

<br />

<h3 id="toc4"><a name="x-Triads-16:18:21, degrees 0, 4, 9."></a>16:18:21, degrees 0, 4, 9.</h3>

<h3 id="toc4"><a name="x-Triads-16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19)."></a>16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).</h3>

18/16 = 9/8 (203.910, error +4.786)<br />

21/16 (470.781, error -1.216)<br />

21/18 = 7/6 (266.871, error: -6.001)<br />

<br />

<h3 id="toc5"><a name="x-Triads-16:18:23, degrees 0, 4, 12."></a>16:18:23, degrees 0, 4, 12.</h3>

<h3 id="toc5"><a name="x-Triads-16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19)."></a>16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).</h3>

18/16 = 9/8 (203.910, error +4.786)<br />

23/16 (628.274, error -2.187)<br />

23/18 (424.364, error: -6.973)<br />

<br />

<h3 id="toc6"><a name="x-Triads-16:21:23, degrees 0, 9, 12."></a>16:21:23, degrees 0, 9, 12.</h3>

<h3 id="toc6"><a name="x-Triads-16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14)."></a>16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).</h3>

21/16 (470.781, error -1.216)<br />

23/16 (628.274, error -2.187)<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc7"><a name="x-Triads-17:18:21, degrees 0, 2, 7."></a>17:18:21, degrees 0, 2, 7.</h3>

<h3 id="toc7"><a name="x-Triads-17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21)."></a>17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).</h3>

18/17 (98.955, error: +5.393)<br />

21/17 (365.825, error: -.608)<br />

21/18 = 7/6 (266.871, error: -6.001)<br />

<br />

<h3 id="toc8"><a name="x-Triads-17:18:23, degrees 0, 2, 10."></a>17:18:23, degrees 0, 2, 10.</h3>

<h3 id="toc8"><a name="x-Triads-17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21)."></a>17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).</h3>

18/17 (98.955, error: +5.393)<br />

23/17 (523.319, error: -1.578)<br />

23/18 (424.364, error: -6.973)<br />

<br />

<h3 id="toc9"><a name="x-Triads-17:21:23, degrees 0, 7, 10."></a>17:21:23, degrees 0, 7, 10.</h3>

<h3 id="toc9"><a name="x-Triads-17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16)."></a>17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).</h3>

21/17 (365.825, error: -.608)<br />

23/17 (523.319, error: -1.578)<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc10"><a name="x-Triads-18:21:23, degrees 0, 5, 8."></a>18:21:23, degrees 0, 5, 8.</h3>

<h3 id="toc10"><a name="x-Triads-18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18)."></a>18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).</h3>

21/18 = 7/6 (266.871, error: -6.001)<br />

23/18 (424.364, error: -6.973)<br />

Line 335:

<h2 id="toc11"><a name="x-Tetrads"></a>Tetrads</h2>

<br />

<h3 id="toc12"><a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9."></a>16:17:18:21, degrees 0, 2, 4, 9.</h3>

<h3 id="toc12"><a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)."></a>16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).</h3>

17/16 (104.955, error -.607)<br />

18/16 = 9/8 (203.910, error +4.786)<br />

Line 343:

21/18 = 7/6 (266.871, error: -6.001)<br />

<br />

<h3 id="toc13"><a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12."></a>16:17:18:23, degrees 0, 2, 4, 12.</h3>

<h3 id="toc13"><a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21)."></a>16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).</h3>

17/16 (104.955, error -.607)<br />

18/16 = 9/8 (203.910, error +4.786)<br />

Line 351:

23/18 (424.364, error: -6.973)<br />

<br />

<h3 id="toc14"><a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12."></a>16:17:21:23, degrees 0, 2, 9, 12.</h3>

<h3 id="toc14"><a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21)."></a>16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).</h3>

17/16 (104.955, error -.607)<br />

21/16 (470.781, error -1.216)<br />

Line 359:

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc15"><a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12."></a>16:18:21:23, degrees 0, 4, 9, 12.</h3>

<h3 id="toc15"><a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19)."></a>16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).</h3>

18/16 = 9/8 (203.910, error +4.786)<br />

21/16 (470.781, error -1.216)<br />

Line 367:

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc16"><a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10."></a>17:18:21:23, degrees 0, 2, 7, 10.</h3>

<h3 id="toc16"><a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21)."></a>17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).</h3>

18/17 (98.955, error: +5.393)<br />

21/17 (365.825, error: -.608)<br />

Line 377:

<h2 id="toc17"><a name="x-Quintad"></a>Quintad</h2>

<br />

<h3 id="toc18"><a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12."></a>16:17:18:21:23, degrees 0, 2, 4, 9, 12.</h3>

<h3 id="toc18"><a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)."></a>16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).</h3>

17/16 (104.955, error -.607)<br />

18/16 = 9/8 (203.910, error +4.786)<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-30 13:11:45 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-30 13:25:49 UTC</tt>.<br>
-: The original revision id was <tt>79928729</tt>.<br>
+: The original revision id was <tt>79930653</tt>.<br>
-: The revision comment was: <tt>listed 10 triads, 6 tetrads, & one quintad in 23edo</tt><br>
+: The revision comment was: <tt>added inversions of chords</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 24: / Line 24: @@
 I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp; 23, excluding (at least for now) 33, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.
-Thus we produce ten triads, five tetrads, &amp; one quintad.
+Thus we produce ten triads, five tetrads, &amp; one quintad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.
 ==Triads==
-===16:17:18, degrees 0, 2, 4.===
+===16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).===
 /16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
 /17 (98.955, error: +5.393)
-===16:17:21, degrees 0, 2, 9.===
+===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).===
 /16 (104.955, error -.607)
 /16 (470.781, error -1.216)
 /17 (365.825, error: -.608)
-===16:17:23, degrees 0, 2, 12.===
+===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).===
 /16 (104.955, error -.607)
 /16 (628.274, error -2.187)
 /17 (523.319, error: -1.578)
-===16:18:21, degrees 0, 4, 9.===
+===16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).===
 /16 = 9/8 (203.910, error +4.786)
 /16 (470.781, error -1.216)
 /18 = 7/6 (266.871, error: -6.001)
-===16:18:23, degrees 0, 4, 12.===
+===16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).===
 /16 = 9/8 (203.910, error +4.786)
 /16 (628.274, error -2.187)
 /18 (424.364, error: -6.973)
-===16:21:23, degrees 0, 9, 12.===
+===16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).===
 /16 (470.781, error -1.216)
 /16 (628.274, error -2.187)
 /21 (157.493, error: -.971)
-===17:18:21, degrees 0, 2, 7.===
+===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).===
 /17 (98.955, error: +5.393)
 /17 (365.825, error: -.608)
 /18 = 7/6 (266.871, error: -6.001)
-===17:18:23, degrees 0, 2, 10.===
+===17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).===
 /17 (98.955, error: +5.393)
 /17 (523.319, error: -1.578)
 /18 (424.364, error: -6.973)
-===17:21:23, degrees 0, 7, 10.===
+===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).===
 /17 (365.825, error: -.608)
 /17 (523.319, error: -1.578)
 /21 (157.493, error: -.971)
-===18:21:23, degrees 0, 5, 8.===
+===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).===
 /18 = 7/6 (266.871, error: -6.001)
 /18 (424.364, error: -6.973)
@@ Line 80: / Line 80: @@
 ==Tetrads==
-===16:17:18:21, degrees 0, 2, 4, 9.===
+===16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).===
 /16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
@@ Line 88: / Line 88: @@
 /18 = 7/6 (266.871, error: -6.001)
-===16:17:18:23, degrees 0, 2, 4, 12.===
+===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).===
 /16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
@@ Line 96: / Line 96: @@
 /18 (424.364, error: -6.973)
-===16:17:21:23, degrees 0, 2, 9, 12.===
+===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).===
 /16 (104.955, error -.607)
 /16 (470.781, error -1.216)
@@ Line 104: / Line 104: @@
 /21 (157.493, error: -.971)
-===16:18:21:23, degrees 0, 4, 9, 12.===
+===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).===
 /16 = 9/8 (203.910, error +4.786)
 /16 (470.781, error -1.216)
@@ Line 112: / Line 112: @@
 /21 (157.493, error: -.971)
-===17:18:21:23, degrees 0, 2, 7, 10.===
+===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).===
 /17 (98.955, error: +5.393)
 /17 (365.825, error: -.608)
@@ Line 122: / Line 122: @@
 ==Quintad==
-===16:17:18:21:23, degrees 0, 2, 4, 9, 12.===
+===16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).===
 /16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
@@ Line 279: / Line 279: @@
 I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp;amp; 23, excluding (at least for now) 33, 55, 79, &amp;amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.&lt;br /&gt;
 &lt;br /&gt;
-Thus we produce ten triads, five tetrads, &amp;amp; one quintad.&lt;br /&gt;
+Thus we produce ten triads, five tetrads, &amp;amp; one quintad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp;amp; I recommend trying different voicings. Moving chord tones up &amp;amp; down by octaves, you can unmuddy a muddy chord.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Triads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Triads&lt;/h2&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Triads-16:17:18, degrees 0, 2, 4."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;16:17:18, degrees 0, 2, 4.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Triads-16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Triads-16:17:21, degrees 0, 2, 9."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;16:17:21, degrees 0, 2, 9.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Triads-16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /17 (365.825, error: -.608)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Triads-16:17:23, degrees 0, 2, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;16:17:23, degrees 0, 2, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Triads-16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Triads-16:18:21, degrees 0, 4, 9."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;16:18:21, degrees 0, 4, 9.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Triads-16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).&lt;/h3&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Triads-16:18:23, degrees 0, 4, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;16:18:23, degrees 0, 4, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Triads-16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).&lt;/h3&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Triads-16:21:23, degrees 0, 9, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;16:21:23, degrees 0, 9, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Triads-16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).&lt;/h3&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
 /21 (157.493, error: -.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Triads-17:18:21, degrees 0, 2, 7."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;17:18:21, degrees 0, 2, 7.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Triads-17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).&lt;/h3&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
 /17 (365.825, error: -.608)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Triads-17:18:23, degrees 0, 2, 10."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17:18:23, degrees 0, 2, 10.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Triads-17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).&lt;/h3&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Triads-17:21:23, degrees 0, 7, 10."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;17:21:23, degrees 0, 7, 10.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Triads-17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).&lt;/h3&gt;
 /17 (365.825, error: -.608)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 /21 (157.493, error: -.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-Triads-18:21:23, degrees 0, 5, 8."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;18:21:23, degrees 0, 5, 8.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-Triads-18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).&lt;/h3&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
@@ Line 335: / Line 335: @@
 &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x-Tetrads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Tetrads&lt;/h2&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;16:17:18:21, degrees 0, 2, 4, 9.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
@@ Line 343: / Line 343: @@
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;16:17:18:23, degrees 0, 2, 4, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
@@ Line 351: / Line 351: @@
 /18 (424.364, error: -6.973)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;16:17:21:23, degrees 0, 2, 9, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
@@ Line 359: / Line 359: @@
 /21 (157.493, error: -.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;16:18:21:23, degrees 0, 4, 9, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).&lt;/h3&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
@@ Line 367: / Line 367: @@
 /21 (157.493, error: -.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;17:18:21:23, degrees 0, 2, 7, 10.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).&lt;/h3&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
 /17 (365.825, error: -.608)&lt;br /&gt;
@@ Line 377: / Line 377: @@
 &lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="x-Quintad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Quintad&lt;/h2&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;16:17:18:21:23, degrees 0, 2, 4, 9, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).&lt;/h3&gt;
 /16 (104.955, error -.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;